Forecasting method of multidimensional time series based on
Neuro-Fuzzy Cognitive Temporal Models
Vadim Borisova and Victor Luferova
a
Branch of the “National Research University “Moscow Power Engineering Institute”
in Smolensk, Smolensk, Russia
Abstract
In the article there are Neuro-Fuzzy Cognitive Temporal Models (NFCTM) described. Those
provide accounting of indirect and indirect mutual impact of all the multidimensional time
series (MTS) components with their temporary delays relative to each other and are oriented
on forecasting of multidimensional time series. Neuro-Fuzzy Cognitive Temporal
Componental Models, which provide the formation of forecasted values of the MTS
components with the temporary delays demanded, are used in NFCTM concepts in order to
accomplish the temporal transformation. There is the way of NFCTM coordinated training
described, which consists in Neuro-Fuzzy Componental Temporary Models for each of the
NFCTM component and also in coherence of these Neuro-Fuzzy Componental Temporary
Models (NFCTM) between each other. There is an MTS forecasting method offered in
condition of unreliability the nonlinearity of the interaction, partial inconsistency and
interdependence of the MTS components, that is based on NFCTM. There are experimental
studies conducted and the results of using the proposed method are presented on the example
of the problem of multidimensional forecasting of the state of the urban environment in
Moscow. The use of the proposed method may be in demand to provide reliable forecasting of
the state of the urban environment in various regions of Russia and other countries, including
into account the complex epidemiological situation.
Keywords1
multidimensional time series, Neuro-Fuzzy Cognitive Temporal Model, Neuro-Fuzzy
Componental Temporal Model.
1. Introduction
Methods based on random process theory, mathematical statistics, and pattern recognition are used
to predict multidimensional time series (MTS). At the same time, as a rule, they are based on approaches
to forecasting of one-dimensional time series and do not fully take into account the non-linear nature of
interaction between the components of the MTS, different quality, insufficient volume and incomplete
information [1-3].
Currently, neural network and fuzzy methods are well established to solve these problems [4, 5], the
limitations of which are the difficulty of taking into account the indirect interplay of MTS components
and their partial coherency.
The multi-criteria nature of analysis and forecasting requires minimization of prediction errors for
all MTS components at the same time. However, this is generally impossible to achieve for complex
systems and processes in real-world conditions of uncertainty, non-linearity of interaction, partial
inconsistency and substantial interdependence of TDM components.
Fuzzy cognitive maps and prediction methods based on them are aimed at solving such problems
[8-10]. However, their use is also limited by the insufficient capacity of the system dynamics models
Russian Advances in Artificial Intelligence: selected contributions to the Russian Conference on Artificial intelligence (RCAI 2020), October
10-16, 2020, Moscow, Russia
EMAIL: vbor67@mail.ru (V. Borisov); lyferov@yandex.ru (V. Luferov)
ORCID: 0000-0001-7357-9365 (V. Borisov); 0000-0002-2499-6135 (V. Luferov)
© 2020 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
used and the lack of consideration of the different time delays of the interdependent components of the
MTS.
The article deals with Neuro-Fuzzy Cognitive Temporal Models (NFCTM) which provide direct
and indirect interaction of all components of multidimensional time series (MTS) with their time delays
relative to each other, and are focused on predicting multidimensional time series. The method of
coordinated training of NFCTM is described, which consists in training of Neuro-Fuzzy Component
Temporal Models for each NFCTM concept, as well as in matching of these Neuro-Fuzzy Component
Temporal Models of NFCTM.
There are experimental studies conducted and the results of using the proposed method are presented
on the example of the problem of multidimensional forecasting of the state of the urban environment in
Moscow. The use of the proposed method may be in demand to provide reliable forecasting of the state
of the urban environment in various regions of Russia and other countries, including into account the
complex epidemiological situation.
2. Neuro-Fuzzy Cognitive Temporal Models for predicting multidimensional
time series
Let’s present the MTS as follows:
S = ( S1 .. S N ) ,
(t )
(
( t −1) ( t − L1 ) )
) ( )
( t − LN )
.. ϕ1, N sN( ) .. sN 1 ,
1
t −1
s1 = F1 ϕ1,1 s1 .. s1
(1)
∀t ∈ {1.. τ ..} St = ...
( ) ( )
sN(t ) = FN ϕ N ,1 s1(t −1) .. s1( N ) .. ϕ N , N sN(t −1) , .. sN( N ) ,
t − L1 t − LN
where S – multidimensional time series; St = ( s1(t ) .. sN(t ) ) – time «slice» of the MTS at the t-th instant of
time; s (jt ) – the value of the j-th component of the MTS at the t-th instant of time; Lij – maximum time
delay of the j-th component of the MTS relative to the i-th; ϕi , j – operator for accounting for the
(t )
interaction between the j-th and i-th MTS components; Fi – transformation for definition si ,
i = 1, ..., N , N – quantity of the MTS components.
Article [9] proposes a new type of NFCTM focused on MTS forecasting:
FCTM = С , W ,
{Ci | i ∈ 1.. N } , N =
C= C,
(t ) i s′(t −1) .. s′(t − Li ) | j ∈=
j
Ci : s i
= F j j 1.. N i , i 1.. N ,
=W {W | i, j ∈1.. N } ,
ij
(2)
{( ) }
j
=Wij ijt − li | l j 0.. Lj ,
w
= i i
(t −lij ) (t −lij ) (t −lij ) j
=s ′ j ϕ=
ij wij , s j , li 0.. Lij ,
where С – multiple NFCTM concepts corresponding to MTS components; Fi – fuzzy temporal
(t )
transformation implemented by the concept Ci ; N – number of NFCTM concepts; s i – predicted fuzzy
value of the concept C at the t-th instant of time; s ′(t −1) .. s ′( ) – subset of the input temporal fuzzy
j
t−L i
i j j
variables of the concept Ci , associated with the corresponding output temporal fuzzy variables of the
concept Cj ;
N i – number of NFCTM concepts directly related to the concept Ci ; li j – time delay for the
corresponding input variable s ′j( ) of the concept C , l j = 0.. Lj ; W – a set of fuzzy degrees of direct
t − li j
i i i
(ijt −l ) of the
j
impact between all pairs of NFCTM concepts; Wij – a subset of fuzzy degrees of impact w
i
concept C j on the concept Ci taking into account the time delay li j ; ϕ ij – fuzzy operator accounting
for the degree of impact of the output variable of the concept C j on the concept's input variable Ci .
3. Description of the method for predicting multidimensional time series
based on Neuro-Fuzzy Cognitive Temporal Models
The method of prediction of MTS based on NFCTM consists of the stages discussed below.
Stage 1. Identification of meaningful components of the MTS for determining the composition of
NFCTM concepts.
The implementation of the proposed method will be considered on the example of multidimensional
forecasting of the state of the urban environment in Moscow. The state of the urban environment is
characterized by the state of its facilities, systems and infrastructure and cannot be reduced to any single
indicator. Basing on the results of previous studies [10-12], the following meaningful factors
(components of MTS) characterizing the state of the urban environment have been determined:
• C1 – ecology of the urban environment;
• C2 – capacity of urban environment infrastructure;
• C3 – income level of the population;
• C4 – industrial consumption of fuel and energy resources;
• C5 – life quality of the population;
• C6 – sanitary and epidemiologic situation.
Stage 2. Determining the fuzzy degrees of impact of the components of the MTS for different time
delays and forming the structure of the NFCTM.
(ijt −l ) taking into account time delays l j for NFCTM
j
To determine the degree of mutual impact w
i
i
concepts, various methods of data analysis can be used, based on the establishment of interdependencies
between all the components of the MTS. For example, for the example under consideration (due to the
different quality of the urban environment, the expert nature of their assessment, the non-linear
relationship between them and the non-stochastic uncertainty), a fuzzy extension of the multiple linear
regression method has been chosen [13].
In table 1 shows the formed matrix of fuzzy relations W of impact of concept sources on concept
receivers of NFCTM for solved task of multidimensional forecasting of urban environment state. For
clarity, only modal values of fuzzy degrees of impact are shown.
Table 1
Formed matrix of fuzzy impact relationships between NFCTM concepts
W li j C1 C2 C3 C4 C5 C6
1 0 0,75 0 0,52 0 0
C1 2 0 0,84 0 0 0 0
3 0 0,40 0 0,40 0 0
1 0 0 0,79 1,0 0 0
C2 2 0 0 0 0 0 0
3 0 0 0 0 0,52 0,57
1 0,55 0 0,68 0,50 0,40 0,43
C3 2 0 1,0 0 0,46 0 0
3 0,61 0 0 0,88 0,99 0
1 0 0,48 0,67 0,79 0 0
C4 2 0 0,41 0 0,43 0 0
3 0,41 0,40 0 0,54 0,49 0
1 0 0,68 0,62 0,42 0,45 1,00
C5 2 0 0,40 0 0 0,48 0
3 1,0 1,00 1,00 0,47 1,00 0,54
1 0 0 0 0 0,53 0,59
C6 2 0 0 0 0 0,51 0
3 0 0 0 0 0 0
Formation of NFCTM structure consists in definition of structural relationships between NFCTM
(ijt −l ) . The formed structure of the NFCTM for multidimensional
j
concepts weighted by fuzzy values w
i
forecasting of the state of the urban environment of Moscow is shown in figure 1.
(t )
s 2 : s2(t )
(t )
(31t −3)
w s 3 : s3(t )
(t −1)
w31
(t − 2)
32
w
(23t −1)
w
С2 С3
(t −1)
w42
(t )
s1 : s1( )
t (42t − 2)
w
(t − 2) ( t − 3)
12
w (24t −1)
w w 34
(42t −3)
w
С1 ( t −1) (
34t −1)
( t − 3)
12 (52t −3) (t −3) w43 w
w w 25 (t − 2)
( t −1) w w 34
52
w (t − 2)
(t ) 52
w
s1 ( t −1)
53
w
( t −1)
14 (53t −3)
w
(t −1)
s1 w
FS1
w14
( t − 3)
(t − 2)
s1 С4
(35t −3)
( t −1) w
(26t −3) 35
w
w (36t −1)
w
54 ( t −1)
w
( t − 3) w
( t − 3)
54 (45t −3)
w
11
w (56t −1)
w (t )
s 4 : s4( )
t
(65t −1)
w С5
(65t − 2)
w
(51t −3)
w
(41t −3)
w
(t )
s 5 : s5( )
t
(t )
s : s6(t )
6
Figure 1. Neuro-Fuzzy Cognitive Temporal Model for multidimensional forecasting of the state of
the urban environment in Moscow
As Neuro-Fuzzy Component Temporal Models FSi , that implement fuzzy temporal transformations
i , modified ANFIS models (Adaptive Neuro-Fuzzy Inference System), providing generation of
F
predicted fuzzy values of MTS components with required time delays [9].
The input variables of the model FSi concept Ci are related to the output variables of those concepts
that have a direct impact on the concept Ci . At the same time input variables Ci are «weighted» by
(ijt −l ) :
j
fuzzy degrees of impact w
i
(t −l ) (t −l ) (t −l ) j
s ′ j
j j j
= =wij Т s j , li 0, ..., Lij , (3)
i i i
where T – operation of the t-norm (min-operation).
The output variables of the model FSi of the concept Ci are intended to generate the predicted
values of the i-th MTS component, corresponding to reasonable time delays.
To build models FSi , both expert information about the components of the MVR and experimental data
can be used. Next, we will consider a mixed version, when the model's rule base is formed by an expert,
and its training is carried out on the basis of a training sample. Let's consider this particular case as an
example of building the structure (and later parametric configuration) of a Neuro-Fuzzy Component
( t −1) ( t − 3) ( t − 3) ( t − 3) ( t − 3)
Temporal Model FS1 . The input variables of the model FS1 – S1′ = s′3 , s′3 , s ′4 , s ′5 , s ′1 , the
output variables of this model – S1 = s1 , s1 , s1 { (t ) ( t −1) (t − 2)
}.
Here is an example of one fuzzy production rule of the model FS1 for the concept C1 of NFCTM:
(t −1) ( t − 3) ( t − 3)
If s ′1 is L AND s ′3 is L AND s ′4 is M
( t − 3) ( t − 3)
AND s ′5 is M AND s ′1 is H , (4)
(t )
( ( t −1)
)
(t − 2)
Then s1 is M AND s1 is M AND s1 is L , ( ) ( )
where L , M , H – fuzzy sets of prerequisites and conclusions of model rules FS1 .
Figure 2 shows an example of a Neuro-Fuzzy Component Temporal Model FS1 .
( t −1)
µ L s ′1
s ′1
( t −1)
µ M s ′1
( t −1)
α1
( )(t )
µ L s1
( t −1)
µ H s ′1
( )
(t )
αp (t ) s1
... µ M s1 Z0
( t − 3) ( t −1)
µ L s ′1 s1
Z−1
s ′1
( t − 3)
µ M s ′1
( t − 3)
αP
( )(t )
µ H s1
Z−1
(t − 2)
s1
( t − 3)
µ H s ′1
Figure 2. Neuro-Fuzzy Component Temporal Model
The model FS1 consists of the following layers of elements.
Layer 1. Layer elements are used to determine the degrees of truth for input variable values relative
to the corresponding fuzzy statements of the prerequisites of all model rules. For p-th rule (p = 1, …,
P) models:
( t −1) ( t −1) (t −3) ′(t −3) (t −3) ′(t −3)
µ L s ′1 = s ′1 ∧ L , µ L s ′3 = s 3 ∧ L, µ M s ′4 = s 4 ∧ M,
(5)
(t −3) ′(t −3) (t −3) ′(t −3)
µ M s ′5 = s 5 ∧ M , µ H s ′1 = s 1 ∧ M.
Layer 2. Layer elements aggregate the truth degrees of rule prerequisites. For p-th rule:
( t −1) ′(t −3) ′(t −3) ′(t −3) ′(t −3)
α p = min µ L s ′1 , µ L s 3 , µ M s 4 , µ M s 5 , µ H s 1 . (6)
Layer 3. Layer elements activate rule conclusions according to the degrees of truth of their
prerequisites based on the implication operation (here, Mamdani implication). For the considered rule:
( )
µ M s1 = min (α p , M ) .
(t )
(7)
Layer 4. The layer element performs the max-disjunction operation, accumulating the activated
conclusions of all the model rules:
(t ) (t )
( ( ) (t ) (t )
s1 = max µ L s1 , ..., µ M s1 , ..., µ H s1 . ( ) ( )) (8)
Layer 5. Layer elements are designed to normalize and output model output variable values with
required time delays:
=
(t ) 0 (t )
s1 ( norm ) Z=
( t −1) −1 ( t )
s1 , s1 ( norm ) Z= ( )
(t − 2) ( t −1)
s1 , s1 ( norm ) Z −1 s1 . ( ) ( ) (9)
Next, we use the notation si( ) for normalized values si(, norm
) t t
.
Value of the output fuzzy variable si(t ) of the model FSi if necessary, is defuzzified using the «center
of gravity» method [14]:
∑ ( s( ) ⋅ µ ( s( ) ) )
M
t t
( ) ( ),
i, m ()
t i, m
(t ) si (t )
=si(t ) def
= s i
m =1
, M Supp s i
=
∑ µ ( s( ) )
M
()
si
t
t
i, m (10)
m =1
=
(t )
s i {=
( µ ( s ) / s ) | m 1, ..., M } ,
()
si
t
(t )
i, m
(t )
i, m
( )–
(t )
where si – defuzzified value of the output variable s i of the model FSi in timepoint t; def s i
(t ) (t )
(t )
defuzzification operator using the «center of gravity» method; si , m – m-th the discretized value of a
(t )
variable s i , m = 1, ..., M ; µ s( ) si(,tm)
i
t ( ) – degree of the identity of the variable s (t )
i for the value si(,tm) ;
( ) – variable carrier s .
Supp s i
(t ) (t )
i
Set of values {s | i = 1, ..., N } at the output of the corresponding models {FS | i = 1, ..., N }
(t )
i i
comprehensively characterizes the predicted state of the urban environment at a given time t.
Stage 3. The coordinated training of NFCTM
For coordinated training of NFCTM, a method is proposed comprising the following two
procedures:
firstly, training Neuro-Fuzzy Component Temporal Models for each NFCTM concept;
secondly, matching of Neuro-Fuzzy Component Temperature models with each other.
Training procedure for Neuro-Fuzzy Component Temporal Models FSi is preceded by the
formation of training samples:
(t −1) (t − L ) (k ) | j ∈ 1,..., N , s (t ) (k ) , k =
j
s′j (k ),..., s′j 1, ..., K , (11)
i
i i
′(t −1) (t − L ) (k ) | j ∈ 1,..., N , s (t ) (k )
j
where j s (k ),..., s′ j
i
i i – input and output variable values in k-th example; K
– number of examples in the training sample.
For the models FSi implementing Mamdani’s inference algorithm [14], modal values and blur
degrees of fuzzy sets of prerequisites and rule conclusions are configurable parameters.
Training procedure for all NFCTM models FSi includes the following steps.
Step 1. For each example of the training selection based on the values of input variables
′(t −1) (t − L ) (k ) | j ∈ 1,..., N
j
i(t()cur ) (k ) .
s j (k ),..., s′j i the model FSi calculates the value of the output variable s
i
Step 2. For all examples of teaching sample, the error function is calculated, depending on the
parameters of the model to be configured:
1 K (t )
( )
2
=Ei ∑
K k =1
si (k ) − si(t()cur ) (k ) . (12)
Step 3. In accordance with the learning algorithm (e.g., an error reverse propagation algorithm or a
genetic algorithm), adjustments are made to the parameters to be adjusted.
Steps 1-3 are iteratively repeated, and model training is considered complete when for each of them
the total error does not exceed the set threshold.
Procedure for matching all Neuro-Fuzzy Component Temporal Models FSi , i = 1, ..., N is performed
after their individual training and consists in such change of modal values and degrees of blur of fuzzy
{ }
ij(t −l ) | l j = 0, ..., Lj between concepts of NFCTM, which provides maximum
j
degrees of impact w
i
i i
increase of prediction accuracy of each component of MTS without deterioration of prediction accuracy
of at least one of other components of MTS. This procedure is preceded by a teaching sample consisting
of data for all TDM components:
(t −1) (t − L ) (q) | j ∈ 1,..., N , s (t ) (q) | i =
j
s′j (q ),..., s′j 1, ..., N , q =
1, ..., Q, (13)
i
i i
where Q – Number of examples in this additional teaching sample.
Procedure for matching all NFCTM FSi , i = 1, ..., N consists in the following steps.
Step 1. For each example from a matching training sample based on the values of input variables
(t −1) (t − L ) (q) | j ∈ 1,..., N | i =
j
1, ..., N all the models FSi , i = 1, ..., N calculate the values of
s′j (q ),..., s′j
i
i
(t )
output variables si ( cur ) (q), i = 1, ..., N .
Step 2. For all sample examples for all models FSi , i = 1, ..., N error functions that depend on
{ }
ij(t −l ) | l j = 0, ..., Lj between NFCTM concepts:
j
configurable fuzzy impact parameters w
i
i i
1 Q (t )
( )
2
Ei = ∑
K q =1
si (q ) − si( ()cur ) (q ) , i = 1, ..., N .
t
(14)
Step 3. According to the genetic algorithm used (e.g, [15]) According to the used genetic algorithm,
{
ij(t −l ) | l j = 0, ..., Lj
}
j
adjustment of customizable parameters of fuzzy degrees of impact is performed w
i
i i
between NFCTM concepts thus, to ensure maximum increase in accuracy of forecasting each of the
components of MTS without deterioration of prediction accuracy of at least one of the other MTS
components.
Steps 1-3 are iteratively repeated, and the procedure for matching all NFCTM is considered
successful if the total error for each of these models does not exceed a certain set threshold (For well-
aligned MTS components), or for these models, the Ejworth-Pareto principle will be implemented, [14],
which, in relation to consistent NFCTM training, is expressed in that it is impossible to maximize the
prediction accuracy of any MTS component without deteriorating the prediction accuracy of at least
one of the other MTS components.
Stage 4. MTS forecast based on trained NFCTM.
MTS forecasting is performed based on a trained NFCTM and consists in calculating the values of
output model variables FSi , i = 1, ..., N by the corresponding sets of values of the input variables of these
models that are set each time.
Experiments were carried out and the results of using the proposed method on the example of
multidimensional and forecasting the state of the urban environment in Moscow were obtained. Figure
3 illustrates the results obtained.
Figure 3. Illustration of the results of multidimensional forecasting of the state of the urban
environment of Moscow based on NFCTM
Table 2 presents a comparative assessment of the results of multidimensional forecasting of the state
of the urban environment in Moscow using an artificial neural network (ANN) and the developed
NFCTM. As a comparison, a multilayer perceptron with a hidden layer of 24 neurons was used, which
showed the best among various ANN variants.
The comparative assessment showed that the use of the proposed method based on NFCTM in small
sample conditions allows to increase the accuracy of the forecast of MTS by an average of 10-15%
compared to the best-performing ANN.
Table 2
Comparative evaluation of the multidimensional forecasting results
Forecasting error, MAPE, %
No MTS components
ANN NFCTM
1. Ecology of the urban environment 7,40 6,91
2. The infrastructure power of the urban environment 1,51 0,13
3. Income level of the population 8,72 9,85
4. Industrial consumption of fuel and energy resources 2,35 1,62
5. Population life quality 2,12 0,55
6. Sanitary and epidemiological situation 5,35 5,31
The article describes Neuro-Fuzzy Cognitive Temporal Models focused on predicting
multidimensional time series and providing for the direct and indirect mutual impact of all components
of the MTS with their time delays relative to each other under conditions of uncertainty.
To implement fuzzy temporal transformations of NFCTM concepts, Neuro-Fuzzy Component
Temporal Models are used, which are modified ANFIS-type models, and provide the formation of
predicted values of MTS components with the required time delays.
The proposed method of consistent training of NFCTM is described, which consists, firstly, in
training Neuro-Fuzzy Component Temporal Models for each concept of NFCTM, and secondly, in the
coordination of these NFCTM.
A method for MTS predicting under conditions of uncertainty, non-linearity of mutual impact,
partial inconsistency and interdependence of MTS components, based on NFCTM, has been developed.
Experimental studies are conducted and the results of using the proposed method are presented on
the example of the problem of multidimensional forecasting of the state of the urban environment in
Moscow. A comparative assessment showed that using this method based on NFCTM in small sample
conditions allows to improve the accuracy of the MTS forecast by an average of 10-15% compared to
the best ANN results.
The use of the proposed method may also be in demand to ensure reliable forecasting of the state of
the urban environment in different regions of Russia and other countries, including taking into account
the difficult epidemiological situation.
Acknowledgements
The reported study was funded by RFBR, project number 19-31-90054.
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